SECT. X.] 



OF STEAM NAVIGATION. 



305 



but this may not be the best form, and therefore let A- : 6 : : *- : ~- = the breadth 

 at any point .r, which substituted for b we have 



1-5 v~ b (V r V x r v) z dx 



Its integral is, 



rh" 



the differential of the power. 



n+l 



n+2 



and when x = /?, it is 



-5 v~ b /r(V v)x V x \ . 



VW \ n + l n +2J ' 



1-5 if- b h I r (V v) VA \ 

 r \~ n + I ~ n + 2/ 



If n = 0, the form of the paddle is a rectangle with the same result as before. 



633. If it be a triangle, n 1, and the result is less than for the rectangle, 

 the velocity and area being the same. 



634. If n = i the form is parabolic, and the result is, 



1-5 pg b h (10 r (V -v)- 6V A) 

 I5~r~ 



There we obviously gain advantage, by getting an equal resistance with 

 less breadth, and by this form the resistance to the paddle is least when it 

 strikes the water obliquely as at A, and increases as its action becomes more 



FIG. 28. 



direct. The velocity for the maximum effect is to the velocity of the vessel as 

 2r l-2A:3r::w:V= ^ . ^W, which is less than for square paddles ; if this 



value of V be inserted in the equation, we have - ^| = the power of the 



paddles when they are of a parabolic form, with the depth h, and breadth b. 



If the form of the exterior edge be more rounded than the vertex of the common 



2 Q 



